In our previous work with A. Agrachev (2004-08) we studied controllability issues for the 2D Euler and Navier-Stokes (NS) systems, which describe the evolution of the velocity field for the motion of homogeneous ideal or viscous incompressible fluid on a two-dimensional domain (such as torus, sphere, hemisphere, Riemannian surface). We assumed the system to be controlled by a forcing, which is a linear combination of few functions on the domain (modes) with time-variant coefficients, which are interpreted as controls. Such forcing is called degenerate.

We managed to show that one can choose a small universal set of controlled modes, such that controls, applied to them, are able to provide –approximate controllability for Navier-Stokes/Euler equation as well as global controllability in projections on any finite-dimensional subspace in the space of velocities (or vorticities).